Discrete Euler-Poincaré and Lie-Poisson equations

In this paper, discrete analogues of Euler-Poincaré and Lie-Poisson reduction theory are developed for systems on finite dimensional Lie groups G with Lagrangians L: TG → R that are G-invariant. These discrete equations provide “reduced ” numerical algorithms which manifestly preserve the symplectic structure. The manifold G × G is used as an approximation of TG,and a discrete Langragian L: G × G → R is constructed in such a way that the Ginvariance property is preserved. Reduction by G results in new “variational” principle for the reduced Lagrangian ℓ: G → R, and provides the discrete Euler-Poincaré (DEP) equations. Reconstruction of these equations recovers the discrete Euler-Lagrange equations developed in [MPS 98, WM 97] which are naturally symplectic-momentum algorithms. Furthermore, the solution of the DEP algorithm immediately leads to a discrete Lie-Poisson (DLP) algorithm. It is shown that when G =SO(n), the DEP and DLP algorithms for a particular choice of the discrete Lagrangian L are equivalent to the Moser-